A Variational Model for Data Fitting on Manifolds by Minimizing the Acceleration of a Bézier Curve

Abstract

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We derive a variational model to fit a composite Bézier curve to a set of data points on a Riemannian manifold. The resulting curve is obtained in such a way that its mean squared acceleration is minimal in addition to remaining close the data points. We approximate the acceleration by discretizing the squared second order derivative along the curve. We derive a closed-form, numerically stable and efficient algorithm to compute the gradient of a Bézier curve on manifolds with respect to its control points, expressed as a concatenation of so-called adjoint Jacobi fields. Several examples illustrate the capabilities and validity of this approach both for interpolation and approximation. The examples also illustrate that the approach outperforms previous works tackling this problem.AMS subject classification (2010). 65K10, 65D10, 65D25, 53C22, 49Q99.

Keywords

• Riemannian manifolds
• curve fitting
• composite Bézier curves
• Jacobi fields
• variational models